Root [of] 8 o - Root [of] 3 - Root [of] 2 - (Root [of] 1) - Root [of] 1 - r v n x - t - S - r - e - d - K c - (p) p a - (g) g m S h L b f 2 3 4 5 ...

f

On reducing many cylinders into a single cylinder of square cross-section.
On cubing a cylinder.
* The parallelepiped a b p g is one-eighth of the parallelepiped n b r g, and its cube root is eight times farther from the face of the parallelepiped n b than is the root of the parallelepiped n b r g; that is, the distances of their roots are eightfold one to the other, because the root of 8 is one face of the parallelepiped away, while the root of 1, r g, is distant from that face eight times the face n b.
* The parallelepiped c b K g, being one-quarter of the greater parallelepiped n b r g, has its root at a distance four times the distance of the root of the greater cylinder o L from the face n b, because the greater root is only one face away from itself, while the said lesser one is four such faces distant from that face.
** Because n b is eight times a b, g b is eight times n b; and because n b is four times c b, m b is four times n b and one and a half times d b; S b is ...
Because a b is 1/8 of n b, n b is one eighth of b g; and because c b is 2/8 of n b, n b is 2/8 of b m; and because d b is 3/8 of n b, n b is 3/8 of the space where its root is found; and because e b is 4/8 of n b, n b is 4/8 of b h; and because b r is 5/8 of n b, n b is 5/8 of the space where its root is found; and because s b is 6/8 of b n, b n is six eighths of the space where its root is found; and so, t b being 7/8 of n b, n b is 7/8 of the space where the root of that space is found; and because n b is 8/8 of n b, n b is 8/8 of the space where its root o L is found, that is, the root of the whole cylinder n b r g.
Because the face of the cylinder n b is divided into eighths, and likewise the length of the same cylinder b g is divided into eighths, and in whatever direction the cylinder is divided into its eighths — whether by length or by breadth — those eighths contain equal quantity, the divisions of the sides being parallel, or, I should say, rectangular. Thus, when the eighth n b v L is separated from such a cylinder, or indeed the eighth a b p g, I shall have removed one and the same quantity of that cylinder, by one of the Elements marked f in the margin. And for this reason, drawing the line f o tangent to the corner n of the cylinder, I find that at a distance along the length of the cylinder equal to the length of its face n b — that is, n v — I have found the cube root of eight, which is two, o v and v L. And again, drawing the straight line f r tangent to the corner of the parallelepiped a b p g, which is likewise 1/8 of the cylinder, I have found at r g the cube root of one eighth of the cylinder, which is one; therefore I have cubed that parallelepiped a b p g. And if, by touching the corner n of the whole cylinder with the straight line f o, I intersect the line o L, which is two — that is, the root of 8 — I have touched the end of the face n b, where the division into eight equal parts lies, and by this I have found the root of the cylinder in the first eighth of the whole cylinder divided lengthwise. And if with the line f r I have touched the corner a of the parallelepiped a b p g, having cut the face of the parallelepiped at its eighth, that is, one, I have found the cube root of 1 eight times farther from the face of the cylinder, which is the cube root of 8, because the space g b is eight times the space L b.

Let n b be the first root of o L in the first eighth of the parallelepiped, that is, in v L; and let a b be the last root at the end of the last eighth of the parallelepiped, which will be the root of 1; and thus the spaces, the cuts, the distances and the magnitudes will be proportioned.

r n is 7/8 of r g.

a b goes 8 times into ... and likewise b n goes 8 times into f g, and so entering ... But say it thus: n b is eight times the ...
a b is 1/8 of n b, and n b is 1/8 of b g; c b is 2/8 of n b, and n b goes 4 times into b m and occupies the 4/8 in forming its root; and b d is 3/8 of n b, and n b goes 3 times into ... S and makes 3/8 of b g; and e b is 4/8 of n b, so n b will be 4/8 of ...
** To be cut at the quarter of the parallelepiped, to cut ...